The Gunners are understood to be working on a deal to bring the highly-rated defender to north London.
According to The Athletic, Arsenal have have submitted a bid worth around £30million for the 22-year-old Netherlands international.
The report adds Ajax are seeking £50million for the star, given he has two years remaining on his current deal.
Timber has made 15 appearances for his nation, and netted twice in 34 outings in the Eredvisie this season.
A move to Mikel Arteta's side may disappoint United boss Erik ten Hag, who previously managed him at Ajax.
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The Red Devils have been long-term admirers of the Dutchman, having already tried to sign him.
Man United were set to lure Timber away from Ajax last summer, only for the player to remain at the Eredivisie giants.
And he is not the only one who got away in the last 12 months.
United have previously failed to secure the services of Frenkie de Jong from Barcelona, despite agreeing a fee with the Catalan giants.
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The player was keen to remain at the Camp Nou, and play Champions League football, something United could not offer at the time.
Instead, Casemiro joined the Red Devils, and it remains to be seen whether Ten Hag looks to continue his pursuit in De Jong.
Another player Man United were interested in acquiring was Cody Gakpo, who astonishingly joined bitter rivals Liverpool from PSV Eindhoven.
The Netherlands international forward, registering 21 goals and 15 assists for PSV in the 2021-22 campaign, opted for a move to Anfield over Old Trafford.
Gakpo previously claimed he had held talks with United, and his ex-boss Rudd van Nistelrooy had even tried to persuade the 24-year-old to Man United.
Another target was Darwin Nunez, and yes, another star who ended up joining Jurgen Klopp's side.
Nunez had caught the eye of many during Benfica's Champions League campaign two seasons ago as he scored six goals in 10 appearances.
United representatives managed to arrange a meeting with the Eagles' president Rui Costa, though it did not happen, and Liverpool took advantage and swooped for his services.
This summer, Tottenham are reluctant to sell United's top target Harry Kane and Arsenal are poised to beat them to West Ham midfielder Declan Rice.
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Napoli striker Victor Osimhen is set to sign a new contract with the Serie A champions and so is defender Kim Min-jae, with both players thought to be on the Red Devils' radar.
Should Arsenal get a deal for Timber over the line, it would highlight United's shortcomings in the transfer window since Ten Hag arrived in 2022.